@Article{BazzanellaLanguascoZaccagnini2010,
  Author    = "D. Bazzanella and A. Languasco and A. Zaccagnini",
  Title     = "Primes in logarithmic intervals",
  Journal   = "Trans. Amer. Math. Soc.",
  Volume    = 362,
  Issue     = 5,
  Pages     = "2667--2684",
  Year      = 2010,
  Url       = "\url{http://www.ams.org/tran/2010-362-05/S0002-9947-09-05009-0/home.html}",
  Abstract  = "We give a new estimate for the integral moments
               of primes in short intervals of the type $(p, p + h]$,
               and prove that for every $\lambda > 1/2$ there exists a
               positive proportion of primes $p \le X$ such that the
               interval $(p, p + \lambda \log X]$ contains at least a
               prime number. We improve Cheer and Goldston's result
               (1987) on the size of real numbers $\lambda > 1$ with
               the property that there is a positive proportion of
               integers $m \le X$ such that the interval
               $(m, m + \lambda \log X]$ contains no primes."
}